Complex Analysis and Operator Theory最新文献

We calculate the rate of convergence of the double rational Fourier series for regular, bounded, measurable, and two-variable functions. The rectangular oscillation of the two-variable function is used to quantify this rate. Additionally, we give an approximation of convergence rate of the double rational Fourier series for continuous functions with generalized bounded variation.

In this paper semicommutators of truncated Toeplitz operators on some model space are discussed. We investigate when the product or the semicommutator of two truncated Toeplitz operators is zero, respectively. We also obtain a necessary and sufficient condition for the semicommutator of two truncated Toeplitz operators to be compact.

In this paper, we consider a class of logarithmically convex domains in ({mathbb {C}}^n), called elementary Reinhardt domains, which can be regarded as a natural generalization of Hartogs triangles. The purpose of this paper is twofold. On one hand, we will compute the explicit forms of the Bergman kernel of weighted Hilbert space with radial symbols. On the other hand, by using the expressions of the weighted Bergman kernel, we will show the regularity of the Berezin transform on the elementary Reinhardt domains.

We study a Schwarz–Pick type inequality for the Schur–Agler class (SA(B_{delta })). In our operator theoretical approach, von Neumann’s inequality for a class of generic tuples of (2times 2) matrices plays an important role rather than holomorphy. In fact, the class (S_{2, gen}(B_{Delta })) consisting of functions that satisfy the inequality for those matrices enjoys

Here, (d_{Delta }) is a function defined by a matrix (Delta ) of functions. Later, we focus on the case when (Delta ) is a matrix of holomorphic functions. We use the pseudo-distance (d_{Delta }) to give a sufficient condition on a diagonalizable commuting tuple T acting on (mathbb {C}^2) for (B_{Delta }) to be a complete spectral domain for T. We apply this sufficient condition to generalizing von Neumann’s inequalities studied by Drury (In: Blei RC, Sidney SJ (eds) Banach spaces, harmonic analysis, and probability theory, lecture notes in mathematics, vol 995. Springer, Berlin, pp 14–32, 1983) and by Hartz–Richter–Shalit (Math Z 301:3877–3894, 2022).

In this research, we analyze the existence of nontrivial solution for a class of non-local multivalued elliptic problems on bounded domain with Direchlet boundary condition. The primary techniques employed consist of variational methods for Locally Lipschitz functional applied to fractional Orlicz–Sobolev space. Our main results generalize some recent findings in the literature to non-smooth cases.

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if (beta >0) and the measure (mu ) is a complex Borel measure on the unit disk ({mathbb {D}}), we define the Hankel type operator (K_{mu ,beta }) by

The operator itself has been widely studied when (mu ) is a positive Borel measure supported on the interval [0, 1). We study the boundedness of (K_{mu ,1}) acting on Hardy spaces and the boundedness of (K_{mu ,alpha }), (alpha >1) acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures (mu 's) such that s-Hankel measure is equal to s-Carleson measure.

We construct an infinite dimensional analysis with respect to non-Gaussian measures of fractional Gamma type which we call fractional Gamma noise measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using generalized Appell polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the fractional Gamma measures. Finally, we gives some new properties of the kernels expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions and we construct the so-called fractional Gamma noise Gel’fand triple.

Given a positive Borel measure (mu ) on the one-dimensional Euclidean space (textbf{R}), consider the centered Hardy–Littlewood maximal function (M_mu ) acting on a finite positive Borel measure (nu ) by

where (r_0(x) = inf {r> 0: mu (B(x,r)) > 0}) and B(x, r) denotes the closed ball with centre x and radius (r > 0). In this note, we restrict our attention to Radon measures (mu ) on the positive real line ([0,+infty )). We provide a complete characterization of measures having weak-type asymptotic properties for the centered maximal function. Although we don’t know whether this fact can be extended to measures on the entire real line (textbf{R}), we examine some criteria for the existence of the weak-type asymptotic properties for (M_mu ) on (textbf{R}). We also discuss further properties, and compute the value of the relevant asymptotic quantity for several examples of measures.

We consider a bounded strictly pseudoconvex domain (Omega subset mathbb {C}^{n}) with (C^{2}) boundary. Then, we show that any compact Ahlfors–David regular subset of (partial Omega ) of Hausdorff dimension (beta in (0,2n-1]) contains a peak set E of Hausdorff dimension equal to (beta ).

Let (xi in mathbb {R}), and (rho _iin mathbb {R}) with (0<|rho _i|<1) for (1le ile 3). For an expanding real matrix

and an integer digit set (D={(0,0,0)^t, (1,0,0)^t, (0,1,0)^t, (0,0,1)^t }subset mathbb {Z}^3), let (mu _{M,D}) be the self-affine measure defined by (mu _{M,D}(cdot )=frac{1}{|D|}sum _{din D}mu _{M,D}(M(cdot )-d)). In this paper, we prove that if (rho _1=rho _2), then (L^2(mu _{M,D})) admits an infinite orthogonal set of exponential functions if and only if (|rho _i|=(p_i/q_i)^{frac{1}{r_i}}) for some (p_i,q_i,r_iin mathbb {N}^+) with (gcd (p_i,q_i)=1) and (2|q_i), (i=1,2). In particular, if (rho _1,rho _2,rho _3in {frac{p}{q}:p,qin 2mathbb {Z}+1}) and (rho _1=rho _2), then there exist at most 4 mutually orthogonal exponential functions in (L^2(mu _{M,D})), and the number 4 is the best.